Theorem lpval 20943

Description: The set of limit points of a subset of the base set of a topology. Alternate definition of limit point in [Munkres] p. 97. (Contributed by NM, 10-Feb-2007.) (Revised by Mario Carneiro, 11-Nov-2013.)

Hypothesis

Ref	Expression
lpfval.1	|- X = U. J

Assertion

Ref	Expression
lpval	|- ( ( J e. Top /\ S C_ X ) -> ( ( limPt ` J ) ` S ) = { x | x e. ( ( cls ` J ) ` ( S \ { x } ) ) } )

Distinct variable groups:   x, J   x, S   x, X

Proof of Theorem lpval

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		lpfval.1	. . . . 5 |- X = U. J
2	1	lpfval 20942	. . . 4 |- ( J e. Top -> ( limPt ` J ) = ( y e. ~P X |-> { x | x e. ( ( cls ` J ) ` ( y \ { x } ) ) } ) )
3	2	fveq1d 6193	. . 3 |- ( J e. Top -> ( ( limPt ` J ) ` S ) = ( ( y e. ~P X |-> { x | x e. ( ( cls ` J ) ` ( y \ { x } ) ) } ) ` S ) )
4	3	adantr 481	. 2 |- ( ( J e. Top /\ S C_ X ) -> ( ( limPt ` J ) ` S ) = ( ( y e. ~P X |-> { x | x e. ( ( cls ` J ) ` ( y \ { x } ) ) } ) ` S ) )
5	1	topopn 20711	. . . . 5 |- ( J e. Top -> X e. J )
6		elpw2g 4827	. . . . 5 |- ( X e. J -> ( S e. ~P X <-> S C_ X ) )
7	5, 6	syl 17	. . . 4 |- ( J e. Top -> ( S e. ~P X <-> S C_ X ) )
8	7	biimpar 502	. . 3 |- ( ( J e. Top /\ S C_ X ) -> S e. ~P X )
9	5	adantr 481	. . . 4 |- ( ( J e. Top /\ S C_ X ) -> X e. J )
10		ssdifss 3741	. . . . . 6 |- ( S C_ X -> ( S \ { x } ) C_ X )
11	1	clsss3 20863	. . . . . . 7 |- ( ( J e. Top /\ ( S \ { x } ) C_ X ) -> ( ( cls ` J ) ` ( S \ { x } ) ) C_ X )
12	11	sseld 3602	. . . . . 6 |- ( ( J e. Top /\ ( S \ { x } ) C_ X ) -> ( x e. ( ( cls ` J ) ` ( S \ { x } ) ) -> x e. X ) )
13	10, 12	sylan2 491	. . . . 5 |- ( ( J e. Top /\ S C_ X ) -> ( x e. ( ( cls ` J ) ` ( S \ { x } ) ) -> x e. X ) )
14	13	abssdv 3676	. . . 4 |- ( ( J e. Top /\ S C_ X ) -> { x | x e. ( ( cls ` J ) ` ( S \ { x } ) ) } C_ X )
15	9, 14	ssexd 4805	. . 3 |- ( ( J e. Top /\ S C_ X ) -> { x | x e. ( ( cls ` J ) ` ( S \ { x } ) ) } e. _V )
16		difeq1 3721	. . . . . . 7 |- ( y = S -> ( y \ { x } ) = ( S \ { x } ) )
17	16	fveq2d 6195	. . . . . 6 |- ( y = S -> ( ( cls ` J ) ` ( y \ { x } ) ) = ( ( cls ` J ) ` ( S \ { x } ) ) )
18	17	eleq2d 2687	. . . . 5 |- ( y = S -> ( x e. ( ( cls ` J ) ` ( y \ { x } ) ) <-> x e. ( ( cls ` J ) ` ( S \ { x } ) ) ) )
19	18	abbidv 2741	. . . 4 |- ( y = S -> { x | x e. ( ( cls ` J ) ` ( y \ { x } ) ) } = { x | x e. ( ( cls ` J ) ` ( S \ { x } ) ) } )
20		eqid 2622	. . . 4 |- ( y e. ~P X |-> { x | x e. ( ( cls ` J ) ` ( y \ { x } ) ) } ) = ( y e. ~P X |-> { x | x e. ( ( cls ` J ) ` ( y \ { x } ) ) } )
21	19, 20	fvmptg 6280	. . 3 |- ( ( S e. ~P X /\ { x | x e. ( ( cls ` J ) ` ( S \ { x } ) ) } e. _V ) -> ( ( y e. ~P X |-> { x | x e. ( ( cls ` J ) ` ( y \ { x } ) ) } ) ` S ) = { x | x e. ( ( cls ` J ) ` ( S \ { x } ) ) } )
22	8, 15, 21	syl2anc 693	. 2 |- ( ( J e. Top /\ S C_ X ) -> ( ( y e. ~P X |-> { x | x e. ( ( cls ` J ) ` ( y \ { x } ) ) } ) ` S ) = { x | x e. ( ( cls ` J ) ` ( S \ { x } ) ) } )
23	4, 22	eqtrd 2656	1 |- ( ( J e. Top /\ S C_ X ) -> ( ( limPt ` J ) ` S ) = { x | x e. ( ( cls ` J ) ` ( S \ { x } ) ) } )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   {cab 2608   _Vcvv 3200   \ cdif 3571   C_ wss 3574   ~Pcpw 4158   {csn 4177   U.cuni 4436   |-> cmpt 4729   ` cfv 5888   Topctop 20698   clsccl 20822   limPtclp 20938

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-top 20699  df-cld 20823  df-cls 20825  df-lp 20940

This theorem is referenced by:  islp  20944  lpsscls  20945